Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,19)(11,20,14,16)(12,18,13,17)(21,23,22,24)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,37)(39,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,14)(10,16,19,20)(12,21,13,22)(17,23,18,24)(26,31)(27,34)(33,36)(35,40);(2,5)(6,7)(9,12,15,13)(10,17,19,18)(11,21,14,22)(16,23,20,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,24)(22,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,12)(10,18,19,17)(11,21,14,22)(16,23,20,24);(1,2)(3,6)(4,5)(7,8)(9,14,15,11)(10,16,19,20)(12,21,13,22)(17,24,18,23)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8106397);